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(x+1) (2 + 4x + 6)

2x³ + 6x² + 10x + 6

Step-by-step explanation:

When multiplying polynomials, you need to distribute and multiply each term from the first polynomial to each term in the second polynomial and then simplify by combining like terms.  In this problem, there is a binomial multiplied by a trinomial, so each term needs to be multiplied and combined:

(x + 1)(2x² + 4x + 6)

= x(2x²) + x(4x) + x(6) + 1(2x²) + 1(4x) + 1(6)

= 2x³ + 4x² + 6x + 2x² + 4x + 6

= 2x³ + 6x² + 10x + 6

geometric series converge the common ratio r has magnitude less than one.

which is equivalent to

for the example we need

|2-x| < 1

-1 < 2-x   and 2-x < 1

x < 3 and x > 1

we can write that

answer: 1 < x < 3

1a. common ratio r=250/25=10 > 1     diverges

1b.   r=50/100=1/2 converges

when these converge they converge to

where a is the first term.

answer: 200

1c. r=-1   so it's not the case that |r| < 1.     diverges

2a. x is the common ratio so we converge when |x|< 1

2b.   r=-10x / -2 = 5x.

we converge when

|5x| < 1

|x| < 1/5

2c.   r=3-2x

-1 < 3-2x and 3-2x < 1

-4 < -2x and 2 < 2x

x < 2 and 1 < x

1 < x < 2

2d. [/tex] r=-48x^3/12 = 4x^3[/tex]

|4x^3| < 1

|x^3| < \frac 1 4

|x| < 4^{-1/3}

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step-by-step explanation: